We present observations of the quiet solar photosphere in the Fe I lines at 6302 Å where at least four different spatial locations exhibit upward-directed supersonic flows. These upflows can only be detected in the circular polarization profiles as a double-peaked structure in the blue lobe of both Fe I lines. We have detected cases of either magnetic polarity in the data. The polarization signals associated with the upflows are very weak, which is probably why they had not been seen before in this type of observation. We propose that the observed flows are the signature of aborted convective collapse, similar to the case reported by Bellot Rubio et al. Our data indicate that this phenomenon occurs frequently in the quiet Sun, which means that many magnetic elements (although the fraction is still unknown) are destroyed even before they are formed completely. The spectral signatures of supersonic upflows reported here are probably present in most spectropolarimetric observations of sufficient signal-to-noise ratio and spatial resolution.